# Columbia Geometric Topology Seminar

## Fall 2012

The GT seminar meets on Fridays in Math. 520, at 1:20PM.
Organizer: Walter Neumann.
Other area seminars. Our e-mail list. Archive of previous semesters

### Fall 2012, Spring 2013

Date Speaker Title
September 7 Organizational meeting
September 14/15 No Columbia Seminar: Brock, Canary, Duchin, Futer, Kahn, Rivin, Wise, Wolpert speak at Hunter College Hyperbolic geometry and Teichmuller theory
September 21/22 No Seminar: Baumslag, Bridson, Fine et al. speak at Stevens Institute and Hunter College Group Theory on the Hudson
September 28 Becca Winarski Mapping Class Groups and Covering Spaces
October 5
Paul Siegel 9:40am Room 507
Aaron Smith 1:20pm
Positive Scalar Curvature and Secondary Index Invariants
A higher Riemann--Hilbert correspondence and extensions
October 12 Christian Zickert Thurston's gluing equations for PGL(n,C)
October 19 No Seminar
October 26 Rob Meyerhof Mom 1.5
November 2
Postponed to Spring
Johanna Kutluhan
POSTPONED
POSTPONED
November 9 Joan Birman The curve complex has dead ends
November 16 9:50am Room 507. NOTE TIME! (SGGT seminars are 11am and 1:20pm) Anne Pichon Lipschitz geometry of complex surfaces: analytic invariants and equisingularity
November 23 No Seminar Thanksgiving
November 30 Colin Adams Petal diagrams for knots
December 7 Jessica Purcell Geodesics in compression bodies
December 14 9:50am
December 14 1:20pm
Chris Leininger
Rinat Kashaev
Finite rigid subsets of the curve complex
The quantum beta function and state integrals of Turaev-Viro type on shaped triangulations

## Abstracts.

#### September 28.

Becca Winarski “ Mapping Class Groups and Covering Spaces ”

Abstract: A fundamental question of the study of mapping class groups is finding maps between mapping class groups of different surfaces. Given a covering space of surfaces, one may wish to relate the mapping class groups of the two surfaces. We say a cover of S over X has the Birman-Hilden property if there is a finite index subgroup of the mapping class group of X that injects into the mapping class group of S, modulo the deck transformations. In the 1970s, Birman and Hilden proved that regular covers have the Birman-Hilden property. We extend their work to certain irregular branched covers. As new applications, we prove: (1) Simple covers do not have the Birman-Hilden property and (2) All covers with at most one branch point do have the Birman-Hilden property.

#### October 5, 9:40AM Room 507.

Paul Siegel “Positive Scalar Curvature and Secondary Index Invariants ”

Abstract: The Atiyah-Singer index theorem calculates algebraic invariants of elliptic operators on manifolds in terms of topological data.  One of the more mysterious applications of this theorem is that it provides topological obstructions to the existence of Riemannian metrics on compact spin manifolds whose scalar curvature function is everywhere positive.  In this talk I will explain how a positive scalar curvature metric on a spin manifold determines a "secondary index invariant" in an object called the analytic structure group, and I will show how homological calculations with the analytic structure group can be used to give nice proofs of old and new theorems about positive scalar curvature metrics.

#### October 5, 1:20pm.

Aaron Smith “A higher Riemann--Hilbert correspondence and extensions ”

Abstract: I will discuss the higher Riemann--Hilbert correspondence of the paper, arxiv 0908.2843, and then some ongoing work describing an extension of this correspondence to a setting involving "A_\infty-connections". The main goal of this extension is to incorporate some natural examples into the framework of the higher R--H correspondence, notably the free loop space with its string product.

#### October 12.

Christian Zickert “Thurston's gluing equations for PGL(n,C) ”

Abstract:Thurston's gluing equations are polynomial equations invented by Thurston to explicitly compute hyperbolic structures or, more generally, representations in PGL(2,C). This is done via so called shape coordinates. We generalize the shape coordinates to obtain a parametrization of representations in PGL(n,C). We give applications to quantum topology, and discuss an intriguing duality between the shape coordinates and the Ptolemy coordinates of Garoufalidis-Thurston-Zickert. The shape coordinates and Ptolemy coordinates can be viewed as 3-dimensional analogues of the X- and A-coordinates on higher Teichmuller spaces due to Fock and Goncharov.

#### October 26.

Rob Meyerhof “Mom 1.5 ”

Abstract:This talk will discuss joint work in progress with Robert Haraway and Craig Hodgson. Mom Technology has had considerable success in proving facts about low-volume 1-cusped hyperbolic 3-manifolds. We are attempting to generalize Mom Technology to the case of hyperbolic 3-manifolds with totally geodesic boundary. The generalization of Mom Technology to this setting has several interesting aspects. For example, a classical Mom(2) for a 1-cusped hyperbolic 3-manifold is a certain type of handlebody structure with two 1-handles and two 2-handles. In the case of a totally geodesic 2-holed-torus boundary, this won't work; in fact, the simplest Mom in this setting would have 1 1-handle and 2 2-handles.

#### November 9.

Joan Birman “The curve complex has dead ends”

Abstract: My talk will be about new joint work with Bill Menasco, on the {\it curve complex} $\C$ of as closed surface $S$ of genus \geq 2. This simplicial complex was introduced by Harvey in the 1970's to study mapping class groups of surfaces, (the MCG) acts on $\C$), and developed into a major tool by Mazur and Minsky during the past 10 years. Vertices are homotopy classes of simple closed curves on $S$. Two vertices $X,Y$ are distance 1 apart (and joined by an edge) when their defining curves are disjoint. It's interesting to think about geodesics joining two vertices, when their defining curves intersect many times. The vertex $Y$ is a {\it dead end} with respect to $X$ if there is no extension of geodesics joining $X$ and $Y$, past $Y$, to longer geodesics. My talk is about new pathology in $\C$: myriad examples of dead ends (and double dead ends) in $\C$, the simplest example being one on a surface of genus 2, with $d(X,Y)=3$. A complete picture emerges when we find nasc for $Y$ to be a dead end with respect to $X$ and prove that every dead end in $\C$ has depth 1.

#### November 16, 9:50 AM.

Anne Pichon “Lipschitz geometry of complex surfaces: analytic invariants and equisingularity ”

Abstract: The question of defining a good notion of equisingularity of a reduced hypersurface $\mathfrak{X} \subset \C^n$ along a non singular complex subspace $Y \subset \mathfrak{X}$ in a neighbourhood of a point $0 \in \mathfrak{X}$ has a long history which started in 1965 with works of Zariski. One of the central concepts introduced by Zariski is the algebro-geometric equisingularity, called nowadays Zariski equisingularity, which defines the equisingularity inductively on the codimension of $Y$ in $\mathfrak{X}$ by requiring that the reduced discriminant locus of a suitably general projection $p\colon \mathfrak{X} \to \C^{n-1}$ be itself equisingular along the strata $p(Y)$.

When $Y$ has codimension one in $\mathfrak{X}$, i.e., when dealing wih a family of plane curves transversal to the parameter space $Y$, it is well known that Zariski equisingularity is equivalent to the main notions of equisingularity such as Whitney conditions for the pair $(\mathfrak{X} \setminus Y,Y)$ and topological triviality. However these properties fail to be equivalent in higher codimension.

I will present a recent joint work with Walter Neumann in which we prove that in codimension $2$, for a family of hypersurfaces in $\C^3$ with isolated singularities, Zariski equisingularity is equivalent to the constancy of the family up to bilipschitz semi-algebraic homeomorphism with respect to the outer metric.

#### November 30.

Colin Adams “Petal diagrams for knots ”

Abstract: Knots have traditionally been investigated by considering projections with crossings where two strands of the knot cross one another. Here, we consider multi-crossings (or n-crossings) where n strands of the knot cross at a single point. We show that for each integer \ge 2, every knot has a projection made up entirely of n-crossings, and therefore a minimal n-crossing number c_n(K). We investigate what is known about c_n(K) and then show that for every knot there is an n such that c_n(K) = 1. In fact, every knot has a projection (a certain type of arc index projection) with a single multi-crossing that looks like a daisy as in a petal diagram. We will consider the implications of this.

#### December 7.

Jessica Purcell “Geodesics in compression bodies ”

Abstract: Every knot complement admits a tunnel system: a collection of arcs whose complement is a handlebody. If the knot complement is hyperbolic, and the system consists of a single unknotting tunnel, there is an open question as to whether the tunnel will be isotopic to a geodesic. In this talk, we address the generalization of the question to geometrically finite compression bodies. We conjecture that when there is a single tunnel, that tunnel will be isotopic to a geodesic. However, we show that when there are multiple tunnels, we can select a tunnel system for which the geodesics in their homotopy classes self-intersect. Parts of this work are joint with Marc Lackenby, and parts with Stephan Burton.

#### December 14 9:50am

Chris Leininger “Finite rigid subsets of the curve complex ”

Abstract: The mapping class group of a surface acts on the complex of curves of that surface, which is an infinite diameter, locally infinite simplicial complex. An important result due to Ivanov (and generalized by Korkmaz and Luo) is that any automorphism of the curve complex of a surface is induced by a mapping class. This has been further extended by Irmak, Shackleton, and Behrstock-Margalit. These results say that various qualities of simplicial map of the curve complex to itself (not assumed to be automorphisms) are induced by mapping classes (and hence are indeed automorphisms). In this talk, I'll describe recent work with J. Aramayona in which we construct *finite* subcomplexes for which every simplicial embedding into the curve complex is the restriction of an automorphism of the entire curve complex, and hence is induced by a mapping class.

#### December 14, 1:20pm

Rinat Kashaev “The quantum beta function and state integrals of Turaev-Viro type on shaped triangulations”

Abstract: A shaped triangulation is a finite triangulation of an oriented pseudo three manifold where each tetrahedron carries dihedral angles of an ideal hyperbolic tetrahedron. I will explain a construction which associates to each shaped triangulation a quantum partition function in the form of an absolutely convergent state integral invariant under shaped 3-2 Pachner moves and the shape gauge transformations generated by the total dihedral angles around internal edges through the Neumann-Zagier Poisson bracket. Similarly to Turaev-Viro theory, the state variables live on edges of the triangulation but take their values on the whole real axis. The tetrahedral weight functions enjoy a manifest tetrahedral symmetry. At least for shaped triangulations of closed 3-manifolds, up to normalization, this partition function is conjectured to be the absolute value squared of the partition function of the Teichmueller TQFT. This is similar to the known relationship between the Turaev-Viro and the Witten-Reshetikhin-Turaev invariants of three manifolds. The talk will be based on the joint work with Feng Luo and Grigory Vartanov arXiv:1210.8393.

# Other relevant information.

## Previous semesters:

Spring 2012, Fall 2011, 2010/11, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.

## Our e-mail list.

Announcements for this seminar, as well as for related seminars and events, are sent to the GT seminar mailing list. You can subscribe directly or by contacting Walter Neumann.